\[a \star 1 = a + 1 \pmod{2^n}\] \[a \star (b \star c) = (a \star b) \star (a \star c)\] The period of the function \(a \mapsto 1 \star a\) is a function of \(n\), which we will denote as \(p(n)\). The first few values of \(p(n)\) are \(1, 1, 2, 4, 4, 8, 8, 8, 8, 16, 16, 16, 16, \ldots\) (OEIS A098820), a slow-growing function. \(p\) is provably divergent in the system ZFC + "there exists a rank-into-rank cardinal." Unfortunately, the latter axiom is so strong that the consistency of the system is seriously doubted. Since the divergence of \(p\) has not been proven otherwise, it remains an unsolved problem.